A fast algorithm for the recursive calculation of dominant singular subspaces

摘要

In many engineering applications it is required to compute the dominant subspace of a matrix A of dimension m×n, with m⪢n. Often the matrix A is produced incrementally, so all the columns are not available simultaneously. This problem arises, e.g., in image processing, where each column of the matrix A represents an image of a given sequence leading to a singular value decomposition-based compression [S. Chandrasekaran, B.S. Manjunath, Y.F. Wang, J. Winkeler, H. Zhang, An eigenspace update algorithm for image analysis, Graphical Models and Image Process. 59 (5) (1997) 321–332]. Furthermore, the so-called proper orthogonal decomposition approximation uses the left dominant subspace of a matrix A where a column consists of a time instance of the solution of an evolution equation, e.g., the flow field from a fluid dynamics simulation. Since these flow fields tend to be very large, only a small number can be stored efficiently during the simulation, and therefore an incremental approach is useful [P. Van Dooren, Gramian based model reduction of large-scale dynamical systems, in: Numerical Analysis 1999, Chapman & Hall, CRC Press, London, Boca Raton, FL, 2000, pp. 231–247].In this paper an algorithm for computing an approximation of the left dominant subspace of size k of A∈Rm×n, with k⪡m,n, is proposed requiring at each iteration O(mk+k2) floating point operations. Moreover, the proposed algorithm exhibits a lot of parallelism that can be exploited for a suitable implementation on a parallel computer.